Total variation regularization for recovering the spatial source term in a time-fractional diffusion equation

TL;DR

This work tackles the inverse problem of recovering a space-dependent source in a time-fractional diffusion equation with partial interior data. It replaces traditional $L^2$ penalties with a total variation regularization within a variational optimal control framework, providing existence and stability results for the continuous problem and convergence of a fully discrete scheme based on finite elements in space and a L1-time discretization. A linearized primal-dual algorithm is developed to solve the resulting saddle-point formulation efficiently, with rigorous guidance on step sizes and convergence. Numerical experiments in 1D and 2D demonstrate that TV regularization produces sharp, non-smoothing reconstructions of discontinuous sources and offers robustness to noise, outperforming classical Tikhonov approaches in preserving edges and support.

Abstract

In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation. To deal with the ill-posedness of the problem, we transform the problem into an optimal control problem with total variational (TV) regularization. In contrast to the classical Tikhonov model incorporating $L^2$ penalty terms, the inclusion of a TV term proves advantageous in reconstructing solutions that exhibit discontinuities or piecewise constancy. The control problem is approximated by a fully discrete scheme, and convergence results are provided within this framework. Furthermore, a lineraed primal-dual iterative algorithm is proposed to solve the discrete control model based on an equivalent saddle-point reformulation, and several numerical experiments are presented to demonstrate the efficiency of the algorithm.

Figures (6)

• Figure 6.1: The computed source functions for Example \ref{['examp-1']} with $\alpha=0.3$, $\mu(t)=\cos(2\pi t)$ and $\omega=[2/50,25/50]$. $(a)$: $\delta_{\mathrm{rel}}=1\%$; $(b)$: $\delta_{\mathrm{rel}}=0.1\%$; $(c)$: the absolute error $|f^n-f^*|$.
• Figure 6.2: The computed source functions for Example \ref{['examp-1']} with $\alpha=0.8$, $\mu(t)\equiv 1$ and $\omega=[28/50,48/50]$. $(a)$: $\delta_{\mathrm{rel}}=5\%$; $(b)$: $\delta_{\mathrm{rel}}=0.5\%$; $(c)$: the absolute error $|f^n-f^*|$.
• Figure 6.3: Approximate solution of equation \ref{['prob-forward']} with $\alpha=0.8$ and $\mu(t)=\cos(2\pi t)$(first row)/ $\mu(t)\equiv 1$(second row). The observation regions are located between the black dashed lines, i.e., $\omega=[2/50,25/50]$. The relative noise level $\delta_{\mathrm{rel}}=0.1\%$. $(a)$ and $(d)$: finite element solution $u(f^*)(x,t)$; $(b)$ and $(e)$: approximation solution $u(f^n)(x,t)$; $(c)$ and $(f)$: the absolute error $|u(f^n)(x,t)-u(f^*)(x,t)|$.
• Figure 6.4: The computed source functions for Example \ref{['examp-1']} with $\alpha=0.5$, $\mu(t)=\sin(5\pi t)$, $\omega=[15/50,35/50]$ and $\delta_{\mathrm{rel}}=0.5\%$. $(a)$: Tikhonov solution; $(b)$: TV solution; $(c)$: the absolute error $|f^n-f^*|$.
• Figure 6.5: Approximate solution of equation \ref{['prob-forward']} with $\alpha=0.5$ and $\mu(t)=\sin(5\pi t)$. The observation regions are located between the black dashed lines, i.e., $\omega=[15/50,35/50]$. The relative noise level $\delta_{\mathrm{rel}}=0.1\%$. $(a)$: finite element solution $u(f^*)(x,t)$; $(b)$: Tikhonov solution $u(f^n)(x,t)$$(c)$: the absolute error of Tikhonov solution; $(d)$: TV solution; $(c)$: the absolute error of TV solution.

Theorems & Definitions (19)

• Lemma 2.1
• Definition 2.1
• Proposition 3.1
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1
• Theorem 4.1
• Theorem 4.2
• Lemma 4.2
• Lemma 4.3